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In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree 0 and rank n on a compact Riemann surface X of genus g. In particular, we prove that such moduli spaces are symplectic singularities, in the sense of Beauville [Bea00], and admit a projective symplectic resolution if and only if g = 1 or (g, n) = (2, 2). These results are an application of a recent paper by Bellamy and Schedler [BS16] via the so-called Isosingularity Theorem.

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